关于积分与极限交换的充要条件

The necessary and sufficient condition of the commutation on integral and limit

由几乎处处收敛的几乎处处连续的本性函数序列的无界点集的闭包S∞是零集,结合几乎处处连续的本性函数的积分定义与可积的充要条件,证明几乎处处收敛的几乎处处连续的本性函数序列的积分与极限交换的充要条件是该函数列在S∞上积分等度收敛.

According to that the closure S∞ of the boundless-point-set of convergent almost everywhere continuous essential function sequence was null set, and the necessary and sufficient conditions of the integral, and the integral definition of the almost everywhere continuous essential function, the necessary and sufficient condition of the commutation on integral and limit, which the almost everywhere continuous essential function sequence was almost everywhere convergent, was proved, the necessary and sufficient condition was that the function sequence was equi-convergent integral on S∞ .